Neural coding refers to how sensory and other information is represented in the brain by neurons. Neurons have been observed to coordinate their firing activity in accordance with the phase of a secondary process, including neuronal field oscillations, external sensory stimuli, and body orientation and/or displacement. This specific form of neural coding is referred to as “phase coding”.
Phase coding appears to be a universal coding trait, manifesting across different organisms and processes. For example, cells in the rodent thalamus and postsubiculum are sensitive to head direction and the activity of different neurons maps to different phases of rotation over the entire 360 degree range. Similarly, cells in the posterior parietal cortex of the macaque monkey respond to vestibular stimuli, coding for a mixture of angular position, velocity, and acceleration. In the blowfly, optic lobe neurons display sensitivity to the motion of visual stimuli and the preferred direction changes with the location of the stimulus over the blowfly's visual field. In the rat hippocampus, gamma-frequency activity from different interneuronal groups was shown to be preferential to different phases of the population theta oscillation, with axo-axonic cells discharging just after the peak crest, oriens-lacunosum-moleculare cells activating on the trough, and parvalbumin-expressing basket cells firing on the descending edge of the theta wave.
The nature of phase coding is dynamic, as phase relationships may change over time. One example is the phenomenon of phase precession, which is a phenomenon of cell firing relative to a “place field”. Place fields are receptive fields corresponding to specific locations in an animal's environment. In the hippocampus, neurons are organized according to place fields. Phase precession is described as a shift in the timing of neuronal firing relative to the local theta rhythm, such that the firing occurs at earlier phases of the theta cycle as an animal passes through a cell's place field. Place field ensembles undergo phase precession and are understood to constitute a cognitive map for navigation. Many studies have implicated the importance of phase precession with regard to the formation and recall of cognitive maps.
Phase selectivity and phase precession are related, sharing a common framework that underpins neural phase coding in general. This framework is the product of both cellular and network mechanisms, such as small molecule signaling, synaptic transmission, electric field coupling, and gap junction coupling. From a systems perspective, however, it is not necessary to characterize these mechanisms explicitly. Their transformative effects may be represented by functional blocks, allowing for a high-level description of neural coding without having to consider microscopic properties or connectivity. Conventional measurement of nonlinear system-level properties and therefore neural coding attributes is achieved by identification of Wiener or Volterra kernel models to map the input-output dynamics of the system. Kernel models of neuronal pathways have yielded insight into how motor and sensory pathways code limb movement, direction, auditory and visual information. Input-output relationships in thalamocortical and hippocampal circuits have also been mapped using similar models.
However, the brain is both a complex signal processor and a generator of rhythmic signals internal to the system. As such, the brain cannot be adequately described using a purely generative model (such as a coupled oscillator model without inputs) or a pure input-output map (such as a Volterra kernel model). Despite the success of kernel models, they cannot account for all forms of neural activity. For example, kernel models cannot account for spontaneous activity lacking an extrinsic trigger, and activity possessing indefinite memory, such as intrinsic oscillations. Rhythms in the brain carry information and their role in neural coding and communication cannot be ignored. Certain neural coding phenomena, such as phase precession, rely strictly on the presence of population oscillations. For example, gamma activity (30-80 Hz) phase-coordinated with the theta rhythm (3-10 Hz) occurs during spatial navigation and is implicated in the formation of hippocampal place fields. Super-gamma ripples (100-300 Hz) superimposed on sharp waves arise during slow-wave sleep (<1 Hz) and are purported to be involved in consolidation of memories in the cortex. The rhythmicity of cortical and hippocampal circuits is attributable to interactions amongst coupled network oscillators consisting of independently rhythmic neuronal subpopulations.
The traditional approach to neural modeling is anchored at the cellular realm, with a single neuron serving as the fundamental network unit. Cellular-level models are robust when the specific properties and connectivity of individual neurons within the network can be established beforehand by measurement or deduction. Examples of such models include classical conductance-based models (Hodgkin and Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J Physiol, 117:500-44, 1952), integrate-and-fire models (Brunel and Hakim, Fast global oscillations in networks of integrate-and-fire neurons with low firing rates, Neural Computation, 11:1621-71, 1999), or hybrids thereof (Breen et al., Hybrid integrate-and-fire model of a bursting neuron, Neural Computation, 15:2843-62, 2003). Some models based on lower-order systems like the second-order Fitzhugh-Nagumo (FN) relaxation oscillator (Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys J, 1:445-66, 1961) are formal mathematical reductions of the higher-order conductance-based models (Tuckwell and Rodriguez, Analytical and simulation results for stochastic Fitzhugh-Nagumo neurons and neural networks, J Comp Neurosci, 5:99-113, 1998), and therefore still associate with the dynamics of single neurons.
In the brain, complex rhythms and patterns of activity arise from ensemble interactions between groups of neurons. Relating specific network phenomena to individual cell-level connections or properties for such large, distributed networks is a daunting task. However, it is possible to replicate the activity of neuronal populations using cellular-level models with sufficiently generous computational resources and time. For example, Traub et al. (Model of the origin of rhythmic population oscillations in the hippocampal slice, Science, 243:1319-25, 1989) developed large-scale conductance and compartmental cell models for the CA1 and CA3 regions of the hippocampus, including one model of the CA3 that incorporated 9000 excitatory pyramidal cells and 900 inhibitory interneurons and were able to reproduce theta population rhythms. Others have focused on smaller networks, such as Kudela et al. (Changing excitation and inhibition in simulated neural networks: effects on induced bursting behavior, Biol Cybern, 88:276-85, 2003), who created a model of approximately 100 single-compartment neurons to explore the inhibitory-excitatory balance and its effect on synchronous population bursting.
Nevertheless, it is difficult to associate or duplicate high-level functionality such as cognition or memory with large networks comprised of many individual biophysically-representative neurons. An alternative approach to modeling neuronal populations involves treating a subpopulation or assembly of similarly behaving neurons as a fundamental network unit, in lieu of the single neuron, thereby limiting the demand on computational resources, improving efficiency, and reducing model parameters and structural complexity. From a modeling perspective, it makes sense to group neurons whose activities are coordinated or similar into assemblies that function as units within a larger network. The concept of neuronal functional blocks from a system perspective leads to a sensible means of organizing and parsing the brain with respect to understanding how aspects of cognition, memory, consciousness and pathology emerge from regional rhythms and connectivity (Buzsaki and Draguhn, Neuronal oscillations in cortical networks, Science, 304:1926-9, 2004).
Physiologically, neuronal assemblies are often observed to behave as oscillators with endogenous rhythmicity (Buzsaki et al., Hippocampal network patterns of activity in the mouse, Neurosci, 116:201-11, 2003), and their interactions are the result of ensemble cellular-level connections (Womelsdorf et al., Modulation of neuronal interactions through neuronal synchronization, Science, 316:1609-12, 2007). A network model of neuronal assemblies should therefore possess differential equations capable of generating oscillations, and include high-level connections representative of effective (ensemble) modalities of physiological coupling.
Wilson and Cowan (Excitatory and inhibitory interactions in localized populations of model neurons, Biophys J, 12:1-24, 1972) were amongst the first to derive coupled differential equations specifically catering to the dynamics of neuronal subpopulations. Their model is an abstraction of system level activity, in that the two dynamic variables, I(t) and E(t), pertain to the fraction of inhibitory and excitatory cells active at time t. Distributions were devised to account for thresholds and synapses, resulting in n-modal sigmoidal transfer functions characterizing the population response to excitation. In contrast, the lumped parameter model of Lopes da Silva (Model of brain rhythmic activity: the alpha-rhythm of the thalamus, Kybernetic, 15:27-37, 1974) generates outputs that are physiologically comparable to electroencephalographic (EEG) and local field recordings, and recent versions of the model have been used to investigate the dynamics of epileptic seizures (Wendling et al., Relevance of nonlinear lumped-parameter models in the analysis of depth-EEG epileptic signals, Biol Cybern, 83:367-78, 2000).
Such models are parametric and therefore require specification of a multitude of parameters that are typically measured from the biological system. This is especially true of cellular conductance-based models. Since the respective parameter values are not always accessible or observable, non-parametric methods, such as kernel models, are useful for describing certain dynamic or functional aspects of neural systems since kernel models do not require explicit knowledge of system parameters or internal structure. Instead, only the observed input(s) and output(s) to the system are needed to characterize the system behavior.
Kernel models of neural pathways have yielded insight into neural coding functionality and input-output dynamics; for example, how motor and sensory pathways code limb movement, direction, auditory and visual information (Gamble and DiCaprio, Nonspiking and spiking proprioceptors in the crab: white noise analysis of spiking CB-chordotonal organ afferents, J Neurophysiol, 89:1815-25, 2003). Input-output relationships in thalamocortical and hippocampal circuits have also been mapped using similar models (Frantseva et al., Changes in membrane and synaptic properties of thalamocortical circuitry caused by hydrogen peroxide, J Neurophysiol, 80:1317-26, 1998). The caveat, however, is that kernel-based models cannot reproduce the dynamics of generative systems that produce a time-varying output independent of any input, including intrinsic oscillations or rhythms (Marmarelis, Modeling methodology for nonlinear physiological systems, Ann Biomed Eng, 25:239-51, 1997), which can be classified as infinite memory phenomena.
It is therefore an object at least to provide a novel method of modeling time-dependent electrical dependent activity of a biological system, a novel cognitive rhythm generator, and a novel method of controlling or treating a seizure-like event.